Dendritic growth into undercooled alloy metals

Lattice Boltzmann model for predicting frosting process on surfaces considering wettability

The process of frosting is a multiscale problem, which leads to challenges of proposing accurate numerical methods. In this study, a lattice Boltzmann model for predicting frost formation and growth on surfaces of various wettabilities is proposed based on the heterogeneous nucleation and dendrite growth theories. Three lattice Boltzmann equations are used to calculate the velocity, humidity, and temperature distributions. Furthermore, the heterogeneous nucleation theory and dendrite growth theory are used to construct the equations that govern ice production during the frosting process, so that the surface wettability can be considered. After experimental validation, the model was used in the analysis of frosting behaviors on plates and in microchannels with different wettabilities. The effects of the intrinsic contact angles and roughness on the frost layer properties were evaluated. This study will likely facilitate a better understanding of frosting on the mesoscopic level.

Three-Dimensional Numerical Simulation of Grain Growth during Selective Laser Melting of 316L Stainless Steel

The grain structure of the selective laser melting additive manufactured parts has been shown to be heterogeneous and spatially non-uniform compared to the traditional manufacturing process. However, the complex formation mechanism of these unique grain structures is hard to reveal using the experimental method alone. In this study, we presented a high-fidelity 3D numerical model to address the grain growth mechanisms during the selective laser melting of 316 stainless steel, including two heating modes, i.e., conduction mode and keyhole mode melting. In the numerical model, the powder-scale thermo-fluid dynamics are simulated using the finite volume method with the volume of fluid method. At the same time, the grain structure evolution is sequentially predicted by the cellular automaton method with the predicted temperature field and the as-melted powder bed configuration as input. The simulation results agree well with the experimental data available in the literature. The influence of the process parameters and the keyhole and keyhole-induced void on grain structure formation are addressed in detail. The findings of this study are helpful to the optimization of process parameters for tailoring the microstructure of fabricated parts with expected mechanical properties.

On Directional Dendritic Growth and Primary Spacing—A Review

The primary spacing is intrinsically linked with the mechanical behavior of directionally solidified materials. Because of this relationship, a significant amount of solidification work is reported in the literature, which relates the primary spacing to the process variables. This review provides a comprehensive chronological narrative on the development of the directional dendritic growth problem over the past 85 years. A key focus within this review is detailing the relationship between key solidification parameters, the operating point of the dendrite tip, and the primary spacing. This review critiques the current state of directional dendritic growth and primary spacing modelling, briefly discusses dendritic growth computational and experimental research, and suggests areas for future investigation.

Experimental and modelling studies for solidification of undercooled Ni-Fe-Si alloys

We present experimental results, analytical calculations and phase-field simulations for undercooled Ni-Fe-Si alloy system. Undercooling experiments are performed using flux encapsulation along with in situ measurement of recalescence speed using a high-speed camera followed by microstructural characterization. Dendrite growth calculations are performed using a modified Boettinger, Coriell and Trivedi theory to incorporate constitutional undercooling due to multiple segregating elements and a modified kinetic undercooling term. Phase-field simulations are performed using a multi-component phase-field model to generate dendrites in this alloy. High growth velocities are observed and the analytical calculations are in good agreement with experiments. The microstructure evolution from the phase-field simulations indicates that there is a difference in solute segregation during growth of dendrites. This article is part of the theme issue 'Heterogeneous materials: metastable and non-ergodic internal structures'.

Volume-Averaged Modeling of Multiphase Flow Phenomena during Alloy Solidification

The most recent developments and applications in volume-averaged modeling of solidification processes have been reviewed. Since the last reviews of this topic by Beckermann and co-workers [Applied Mech. Rev. 1993, p. 1; Annual Rev. Heat Transfer 1995, p. 115], major progress in this area has included i) the development of a mixed columnar-equiaxed solidification model; ii) further consideration of moving crystals and crystal dendritic morphology; and iii) the model applications to analyze the formation mechanisms of macrosegregation, as-cast structure, shrinkage cavity and porosity in different casting processes. The capacity of computer hardware is still a limiting factor. However, many calculation examples, as verified by the laboratory casting experiments, or even by the casting processes at a small industrial scale, show great application potential. Following the trend of developments in computer hardware (projection according to Moore’s law), a full 3D calculation of casting at the industry scale with the multiphase volume-averaged solidification models will become practically feasible in the foreseeable future.

Numerical Simulation of Three-Dimensional Mesoscopic Grain Evolution: Model Development, Validation, and Application to Nickel-Based Superalloys

The mesoscopic grain model is a multiscale model which takes into account both the dendrite growth mechanism and the vast numerical computation of the actual castings. Due to the pursuit of efficient computation, the mesoscopic grain calculation accuracy is lower than that of dendrite growth model. Improving the accuracy of mesoscopic grain model is a problem to be solved urgently. In this study, referring to the calculation method of solid fraction in microscopic dendrite growth model, a cellular automata model of 3D mesoscopic grain evolution for solid fraction calculated quantitatively at the scale of cell is developed. The developed model and algorithm validation for grain growth simulation is made by comparing the numerical results with the benchmark experimental data and the analytical predictions. The results show that the 3D grain envelopes simulated by the developed model and algorithm are coincident with the shape predicted by the analytical model to a certain extent. Then, the developed model is applied to the numerical simulation of solidification process of nickel-based superalloys, including equiaxed and columnar dendritic grain growth. Our results show good agreement with the related literature.

Thermo-solutal growth of an anisotropic dendrite with six-fold symmetry

A stable growth of dendritic crystal with the six-fold crystalline anisotropy is analyzed in a binary nonisothermal mixture. A selection criterion representing a relationship between the dendrite tip velocity and its tip diameter is derived on the basis of morphological stability analysis and solvability theory. A complete set of nonlinear equations, consisting of the selection criterion and undercooling balance condition, which determines implicit dependencies of the dendrite tip velocity and tip diameter as functions of the total undercooling, is formulated. Exact analytical solutions of these nonlinear equations are found in a parametric form. Asymptotic solutions describing the crystal growth at small Péclet numbers are determined. Theoretical predictions are compared with experimental data obtained for ice dendrites growing in binary water-ethylenglycol solutions as well as in pure water.

From atomistic interfaces to dendritic patterns

Transport processes around phase interfaces, together with thermodynamic properties and kinetic phenomena, control the formation of dendritic patterns. Using the thermodynamic and kinetic data of phase interfaces obtained on the atomic scale, one can analyse the formation of a single dendrite and the growth of a dendritic ensemble. This is the result of recent progress in theoretical methods and computational algorithms calculated using powerful computer clusters. Great benefits can be attained from the development of micro-, meso- and macro-levels of analysis when investigating the dynamics of interfaces, interpreting experimental data and designing the macrostructure of samples. The review and research articles in this theme issue cover the spectrum of scales (from nano- to macro-length scales) in order to exhibit recently developing trends in the theoretical analysis and computational modelling of dendrite pattern formation. Atomistic modelling, the flow effect on interface dynamics, the transition from diffusion-limited to thermally controlled growth existing at a considerable driving force, two-phase (mushy) layer formation, the growth of eutectic dendrites, the formation of a secondary dendritic network due to coalescence, computational methods, including boundary integral and phase-field methods, and experimental tests for theoretical models-all these themes are highlighted in the present issue.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

Modeling of coupled motion and growth interaction of equiaxed dendritic crystals in a binary alloy during solidification

Motion of growing dendrites is a common phenomenon during solidification but often neglected in numerical simulations because of the complicate underlying multiphysics. Here a phase-field model incorporating dendrite-melt two-phase flow is proposed for simulating the dynamically interacted process. The proposed model circumvents complexity to resolve dendritic growth, natural convection and solid motion simultaneously. Simulations are performed for single and multiple dendritic growth of an Al-based alloy in a gravity environment. Computing results of an isolated dendrite settling down in the convective supersaturated melt shows that solid motion is able to overwhelm solutal convection and causes a rather different growth morphology from the stationary dendrite that considers natural convection alone. The simulated tip growth dynamics are correlated with a modified boundary layer model in the presence of melt flow, which well accounts for the variation of tip velocity with flow direction. Polycrystalline simulations reveal that the motion of dendrites accelerates the occurrence of growth impingement which causes the behaviors of multiple dendrites are distinct from that of single dendrite, including growth dynamics, morphology evolution and movement path. These polycrystalline simulations provide a primary understanding of the sedimentation of crystals and resulting chemical homogeneity in industrial ingots.

Determination and controlling of grain structure of metals after laser incidence: Theoretical approach

There are serious questions about the grain structure of metals after laser melting and the ways that it can be controlled. In this regard, the current paper explains the grain structure of metals after laser melting using a new model based on combination of 3D finite element (FE) and cellular automaton (CA) models validated by experimental observation. Competitive grain growth, relation between heat flows and grain orientation and the effect of laser scanning speed on final micro structure are discussed with details. Grains structure after laser melting is founded to be columnar with a tilt angle toward the direction of the laser movement. Furthermore, this investigation shows that the grain orientation is a function of conduction heat flux at molten pool boundary. Moreover, using the secondary laser heat source (SLHS) as a new approach to control the grain structure during the laser melting is presented. The results proved that the grain structure can be controlled and improved significantly using SLHS. Using SLHS, the grain orientation and uniformity can be change easily. In fact, this method can help us to produce materials with different local mechanical properties during laser processing according to their application requirements.

Stochastic kinetics reveal imperative role of anisotropic interfacial tension to determine morphology and evolution of nucleated droplets in nematogenic films

For isotropic fluids, classical nucleation theory predicts the nucleation rate, barrier height and critical droplet size by ac- counting for the competition between bulk energy and interfacial tension. The nucleation process in liquid crystals is less understood. We numerically investigate nucleation in monolayered nematogenic films using a mesoscopic framework, in par- ticular, we study the morphology and kinetic pathway in spontaneous formation and growth of droplets of the stable phase in the metastable background. The parameter κ that quantifies the anisotropic elastic energy plays a central role in determining the geometric structure of the droplets. Noncircular nematic droplets with homogeneous director orientation are nucleated in a background of supercooled isotropic phase for small κ. For large κ, noncircular droplets with integer topological charge, accompanied by a biaxial ring at the outer surface, are nucleated. The isotropic droplet shape in a superheated nematic background is found to depend on κ in a similar way. Identical growth laws are found in the two cases, although an unusual two-stage mechanism is observed in the nucleation of isotropic droplets. Temporal distributions of successive events indi- cate the relevance of long-ranged elasticity-mediated interactions within the isotropic domains. Implications for a theoretical description of nucleation in anisotropic fluids are discussed.

Analysis of macrosegregation formation and columnar-to-equiaxed transition during solidification of Al-4 wt.%Cu ingot using a 5-phase model

A 5-phase mixed columnar-equiaxed solidification model was recently introduced to predict the as-cast structure, and a series of laboratory experiments were performed previously to verify the model. The focus of the current work is to analyze the formation of macrosegregation, which accompanies the formation of the as-cast structure including the columnar-to-equiaxed transition (CET). The as-cast structure and macrosegregation map of a cylindrical Al-4 wt.% Cu ingot poured at 800 °C are used as a reference to validate the calculations. Good agreement between the calculations and the experiment regarding both the macrosegregation and CET is achieved. Thermal-solutal convection and equiaxed crystal sedimentation in such ingot are verified to be key mechanisms governing the formation of macrosegregation. The competitive equiaxed/columnar growth, the soft and hard blocking mechanisms predominate the CET. The numerical study of grid sensitivity indicates that the global segregation pattern and CET are not significantly affected by grid size; however, some fine details of the segregation map which are predicted by fine grid (~0.5 mm) are smeared or locally averaged by the coarse grid (~2 mm). Such details were also not resolved in the measurement. Future investigations are demanding.

Modeling diffusion-governed solidification of ternary alloys - Part 2: Macroscopic transport phenomena and macrosegregation

Part 1 of this two-part investigation presented a multiphase solidification model incorporating the finite diffusion kinetics and ternary phase diagram with the macroscopic transport phenomena (Wu et al., 2013). In Part 2, the importance of proper treatment of the finite diffusion kinetics in the calculation of macrosegregation is addressed. Calculations for a two-dimensional (2D) square casting (50 × 50 mm²) of Fe-0.45 wt.%C-1.06 wt.%Mn considering thermo-solutal convection and crystal sedimentation are performed. The modeling result indicates that the infinite liquid mixing kinetics as assumed by classical models (e.g., the Gulliver-Scheil or lever rule), which cannot properly consider the solute enrichment of the interdendritic or inter-granular melt at the early stage of solidification, might lead to an erroneous estimation of the macrosegregation. To confirm this statement, further theoretical and experimental evaluations are desired. The pattern and intensity of the flow and crystal sedimentation are dependent on the crystal morphologies (columnar or equiaxed); hence, the potential error of the calculated macrosegregation caused by the assumed growth kinetics depends on the crystal morphology. Finally, an illustrative simulation of an engineering 2.45-ton steel ingot is performed, and the results are compared with experimental results. This example demonstrates the model applicability for engineering castings regarding both the calculation efficiency and functionality.

Simulation of macrosegregation in a 2.45-ton steel ingot using a three-phase mixed columnar-equiaxed model

A three-phase mixed columnar-equiaxed solidification model is used to calculate the macrosegregation in a 2.45 ton steel ingot. The main features of mixed columnar-equiaxed solidification in such an ingot can be quantitatively modelled: growth of columnar dendrite trunks; nucleation, growth and sedimentation of equiaxed crystals; thermosolutal convection of the melt; solute transport by both convection and crystal sedimentation; and the columnar-to-equiaxed transition (CET). The predicted as-cast macrostructure and the segregation pattern are in qualitative agreement with the reported experimental results. Parameter study on the numerical grid size and the nucleation of the equiaxed crystals are performed, and some segregation mechanisms are numerically analyzed. Discontinued positive-negative segregation just below the hot top is predicted because of the formation of a local mini-ingot and the subsequent sedimentation of equiaxed grains within the mini-ingot. Quasi A-segregates in the middle radius region between the casting outer surface and the centreline are also found. The quasi A-segregates originate from the flow instability, but both the appearance of equiaxed crystals and their interaction with the growing columnar dendrite tips significantly strengthen the segregates. The appearance of equiaxed phase is not a necessary condition for the formation of quasi A-segregates. The quantitative discrepancy between the predicted and experimental results is also discussed.

Single crystal growth and anisotropic crystal-fluid interfacial free energy in soft colloidal systems

We measure the anisotropy of the crystal-fluid interfacial free energy in soft colloidal systems. A temperature gradient is used to direct crystal nucleation and control the growth of large single crystals in order to achieve well-equilibrated crystal-fluid interfaces. Confocal microscopy is used to follow both the growth process and the equilibrium crystal-fluid interface at the particle scale: heterogeneous crystal nucleation, the advancing interface, and the stationary equilibrium interface. We use the measured growth velocity to determine the chemical potential difference between crystal and fluid phases. Well-equilibrated, large crystal-fluid interfaces are then used to determine the interfacial free energy and its anisotropy directly from thermally excited interface fluctuations. We find that while the measured average interfacial free energy is in good agreement with values found in simulations, the anisotropy is significantly larger than simulation values. Finally, we investigate the effect of impurities on the advancing interface. We determine the critical force needed to overcome impurity particles from the local interface curvature.

Prediction of the operating point of dendrites growing under coupled thermosolutal control at high growth velocity

We use a phase-field model for the growth of dendrites in dilute binary alloys under coupled thermosolutal control to explore the dependence of the dendrite tip velocity and radius of curvature upon undercooling, Lewis number (ratio of thermal to solutal diffusivity), alloy concentration, and equilibrium partition coefficient. Constructed in the quantitatively valid thin-interface limit, the model uses advanced numerical techniques such as mesh adaptivity, multigrid, and implicit time stepping to solve the nonisothermal alloy solidification problem for material parameters that are realistic for metals. From the velocity and curvature data we estimate the dendrite operating point parameter σ*. We find that σ* is nonconstant and, over a wide parameter space, displays first a local minimum followed by a local maximum as the undercooling is increased. This behavior is contrasted with a similar type of behavior to that predicted by simple marginal stability models to occur in the radius of curvature, on the assumption of constant σ*.

A New Growth Kinetics in Simulation of Dendrite Growth by Cellular Automaton Method

A modified cellular automaton model was proposed to simulate the dendrite growth of alloy. Different from previous models, this model used neither an analytical equation(such as KGT model) nor an interface solute gradient equation to solve the velocity of solid-liquid interface, but used the interface solute and energy conservation and thermodynamic equilibrium condition to describe the solid/liquid interface growth kinetics process. In present model, once the temperature field and solute field were solved by finite different method in the entire domain, the material thermodynamic properties can be substituted into four algebraic equations to easily determine the variation of solid fraction, interface temperature and solute concentration, instead of calculating interface moving velocity. As a result, the complexity of the calculation can be largely reduced. The simulated dendrite growth was in a good agreement with the Lipton–Glicksman–Kurz (LGK) model for free dendritic growth in undercooled melts.